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Physical Review A

Three-mode squeezed vacuum state in Fock space as an entangled state

Hong-Yi Fan, Guichuan Yu

Published Feb 28, 2002• 21 citations
Publisher

Abstract

We show that the operator $U=\mathrm{exp}[\ensuremath{-}{ir(Q}_{1}{P}_{2}{+Q}_{2}{P}_{3}{+Q}_{3}{P}_{1})]$ is a three-mode squeezing operator for the three-mode quadratures exhibiting the standard squeezing, the corresponding squeezed vacuum state in three-mode Fock space is derived by virtue of the technique of integration within ordered product of operators. The entanglement involved in such a state is explained. The optical network for producing the three-mode squeezed state is proposed.

DOI
10.1103/physreva.65.033829

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In a 3-wave mixing process occurring in some nonlinear optical medium when }a1†a_{1}^{\dagger}a1†​{\small mode interacts with both }$a_{2}^{\dagger}% modeand{\small mode and }modeanda_{3}^{\dagger}mode,wetheoreticallystudythesqueezingeffectgeneratedbytheoperator{\small mode, we theoretically study the squeezing effect generated by the operator }mode,wetheoreticallystudythesqueezingeffectgeneratedbytheoperatorS_{3}\equiv \exp[μ(a_{1}% a_{2}-a_{1}^{\dagger}a_{2}^{\dagger})+ν(a_{1}a_{3}-a_{1}^{\dagger}% a_{3}^{\dagger})].Thenew3−modesqueezedvacuumstateinFockspaceisderived,andtheuncertaintyrelationforitisdemonstrated,Itturnsoutthat{\small . The new 3-mode squeezed vacuum state in Fock space is derived, and the uncertainty relation for it is demonstrated, It turns out that }.Thenew3−modesqueezedvacuumstateinFockspaceisderived,andtheuncertaintyrelationforitisdemonstrated,ItturnsoutthatS_{3}mayexhibitenhancedsqueezing.Byvirtueofthetechniqueofintegrationwithinanorderedproduct(IWOP)ofoperators,wealsoderive{\small may exhibit enhanced squeezing. By virtue of the technique of integration within an ordered product (IWOP) of operators, we also derive }mayexhibitenhancedsqueezing.Byvirtueofthetechniqueofintegrationwithinanorderedproduct(IWOP)ofoperators,wealsoderiveS_{3}${\small 's normally ordered expansion. The Wigner function of new 3-mode squeezed vacuum state is calculated by using the Weyl ordering invariance under similar transformations.
We find that the exponential operator V ≡exp[ i λ( Q 1 P 2 + Q 2 P 3 +⋅⋅⋅+ Q n −1 P n + Q n P 1 )], Q i , P i are, respectively, the coordinate and momentum operators, is an n -mode squeezing operator which engenders standard squeezing. By virtue of the technique of integration within an ordered product of operators we derive V 's normally ordered expansion and obtain the n -mode squeezed vacuum states, its Wigner function is calculated by using the Weyl ordering invariance under similar transformations.
We introduce the three-mode entangled state and set up an experiment to generate it. Then we discuss the three-mode squeezing operator squeezed | p , χ 2 , χ 3 ⟩ → μ -3/2 | p /μ, χ 2 /μ, χ 3 /μ ⟩ and the optical implement to realize such a squeezed state. We also reveal that c -number asymmetric shrink transform in the three-mode entangled state, i.e. | p , χ 2 , χ 3 ⟩ → μ -1/2 | p /μ, χ 2 , χ 3 ⟩, maps onto a kind of one-sided three-mode squeezing operator exp {iλ/3 (∑ i =1 3 P i ) (∑ i =1 3 Q i ) − λ/2}. Using the technique of integration within an ordered product (IWOP) of operators, we derive their normally ordered forms and construct the corresponding squeezed states.